On the existence and generation of non- trivial BIBDs

Figures & data

Table 1. Generated BIBDs. The parameter pairs $\left(v,k\right)$ comprise those pairs with $v\le 40$ and $k\le v/2,$ for which non- trivial BIBDs were not known. The lengths of the trivial BIBDs are denoted with $b_t.$ The smallest length satisfying the necessary conditions (1)(1) $$b\ge v$$(1) , (2)(2) $$v r=b k$$(2) , and (3)(3) $$\lambda \left(v-1\right)=r\left(k-1\right).$$(3) is denoted with $b\prime .$ The lengths of the BIBDs generated with two methods are denoted with $b.$ The factor $f$ in parentheses says that $b$ is the $f$- fold of $b\prime .$

$\mathit{v}$	$\mathit{k}$	$\mathit{b}$t	$\mathit{b}$' Smallest admissible	$\mathit{b}$ Recursive method		$\mathit{b}$ Numerical method
26	11	7726160	130	1040	(8)	520	(4)*
26	12	9657700	325	650	(2)	650	(2)*
28	11	21474180	756	19656	(26)	1512	(2)*
28	13	37442160	252	19656	(78)	1008	(4)*
30	11	54627300	870	24360	(28)	3480	(4)*
30	12	86493225	145	4060	(28)	290	(2)*
30	13	119759850	870	24360	(28)	1740	(2)*
30	14	145422675	435	12180	(28)	870	(2)*
33	10	92561040	528	32736	(62)	2112	(4)*
33	13	573166440	264	32736	(124)	1056	(4)*
33	14	818809200	528	32736	(62)	2112	(4)*
33	15	1037158320	176	10912	(62)	704	(4)*
34	10	131128140	187	46376	(248)	1496	(8)*
34	13	927983760	374	371008	(992)	2992	(8)*
34	14	1391975640	561	556512	(992)	4488	(8)*
34	15	1855967520	374	371008	(992)	2992	(8)*
34	16	2203961430	187	185504	(992)	2992	(16)
35	11	417225900	595	147560	(248)	4760	(8)
35	12	834451800	595	590240	(992)	2380	(4)
35	13	1476337800	595	590240	(992)	4760	(8)
35	15	3247943160	119	1298528	(10912)	1904	(16)
35	16	4059928950	595	6492640	(10912)	4760	(8)
36	11	600805296	252	1062432	(4216)	2016	(8)
36	13	2310789600	1260	21248640	(16864)	5040	(4)*
36	17	8597496600	1260	13749120	(10912)	10080	(8)*
38	10	472733756	703	12654	(18)	5624	(8)
38	11	1203322288	1406	50616	(36)	5624	(4)*
38	12	2707475148	703	25308	(36)	5624	(8)
38	13	5414950296	1406	50616	(36)	5624	(4)*
38	14	9669554100	703	25308	(36)	5624	(8)
38	15	15471286560	1406	50616	(36)	11248	(8)*
38	16	22239974430	703	25608	(36)	11248	(16)
38	17	28781143380	1406	50616	(36)	11248	(8)*
39	10	635745396	741	493506	(666)	5928	(8)
39	11	1676056044	741	493506	(666)	5928	(8)
39	12	3910797436	247	329004	(1332)	3952	(16)
39	14	15084504396	741	987012	(1332)	11856	(16)
39	15	25140840660	247	329004	(1332)	7904	(32)
39	16	37711260990	741	987012	(1332)	11856	(16)
39	17	51021117810	741	987012	(1332)	11856	(16)
39	18	62359143990	247	329004	(1332)	7904	(32)
40	11	2311801440	1560	19740240	(12654)	6240	(4)
40	17	88732378800	1560	39480480	(25308)	12480	(8)
40	18	113380261800	260	6580080	(25308)	8320	(32)
40	19	131282408400	520	692640	(1332)	8320	(16)

*Here, BIBDs for the smallest admissible parameters $b\prime$ were generated with the advanced numerical methods. Thus, $b=b\prime$ and $f=1$ was achieved.

Table A1. The construction of BIBDs for $v=26$ and $k=11$ and $12.$ The length of the smallest admissible design is $b\prime .$

k	$b_1$	$b_2$	q	$n_1$	$n_2$	$b$	$b\prime$
11	300	40	$\frac{40\times 15}{300\times 11}=\frac{2}{11}$	2	11	$2\times 300+11\times 40=1040$	$130$
12	50	300	$\frac{300\times 14}{50\times 12}=\frac{7}{1}$	7	1	$7\times 50+1\times 300=650$	$325$